If a, b be two fixed positive integers such that f(a + x) = b + [… - Vidyadip

MCQ

If $a, b$ be two fixed positive integers such that $f(a + x) = b + {[{b^3} + 1 - 3{b^2}f(x) + 3b{\{ f(x)\} ^2} - {\{ f(x)\} ^3}]^{\frac{1}{3}}}$ for all real $x$, then $f(x)$ is a periodic function with period

A
$a$
✓
$2a$
C
$b$
D
$2b$

✓

Answer

Correct option: B.

$2a$

b
(b) $f(a + x) = b + {(1 + {\{ b - f(x)\} ^3})^{1/3}}$
==> $f(a + x) - b = {\{ 1 - {\{ f(x) - b\} ^3}\} ^{1/3}}$
==> $\phi (a + x) = {\{ 1 - {\{ \phi (x)\} ^3}\} ^{1/3}}$, [$\phi (x) = f(x) - b$]
==> $\phi (x + 2a) = {\{ 1 - {\{ \phi (x + a)\} ^3}\} ^{1/3}} = \phi (x)$
==> $f(x + 2a) - b = f(x) - b \Rightarrow f(x + 2a) = f(x)$
$\therefore$ $f(x)$ is periodic with period $2a$.

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